Farnsley was born in Louisville and attended Male High School there. He received undergraduate and law degrees from the University of Louisville and practiced law in Louisville beginning in 1930. Farnsley served in the Kentucky House of Representatives from 1936 to 1940 and as mayor of Louisville from 1948 to 1953. Farnsley was a very popular mayor, known for his accessibility and attention to problems large and small.

In 1964 Farnsley won Kentucky's Third District congressional seat, defeating the Republican incumbent, Marion M. "Gene" Snyder, who later was elected to the House from an adjoining district. Farnsley served in the house from January 3, 1965 through January 3, 1967. He did not seek re-election to the House.